$ D = \left[\begin{array}{rr}1 & -1 \\ 0 & 3\end{array}\right]$ $ C = \left[\begin{array}{rrr}5 & 2 & 5 \\ 2 & 4 & 5\end{array}\right]$ What is $ D C$ ?
Answer: Because $ D$ has dimensions $(2\times2)$ and $ C$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ D C = \left[\begin{array}{rr}{1} & {-1} \\ {0} & {3}\end{array}\right] \left[\begin{array}{rrr}{5} & \color{#DF0030}{2} & \color{#9D38BD}{5} \\ {2} & \color{#DF0030}{4} & \color{#9D38BD}{5}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{1}\cdot{5}+{-1}\cdot{2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{5}+{-1}\cdot{2} & ? & ? \\ {0}\cdot{5}+{3}\cdot{2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{5}+{-1}\cdot{2} & {1}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{4} & ? \\ {0}\cdot{5}+{3}\cdot{2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{1}\cdot{5}+{-1}\cdot{2} & {1}\cdot\color{#DF0030}{2}+{-1}\cdot\color{#DF0030}{4} & {1}\cdot\color{#9D38BD}{5}+{-1}\cdot\color{#9D38BD}{5} \\ {0}\cdot{5}+{3}\cdot{2} & {0}\cdot\color{#DF0030}{2}+{3}\cdot\color{#DF0030}{4} & {0}\cdot\color{#9D38BD}{5}+{3}\cdot\color{#9D38BD}{5}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}3 & -2 & 0 \\ 6 & 12 & 15\end{array}\right] $